Subdivision Surfaces. Basic Idea
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1 Subdvson Surfces COS598b Geometrc Modelng Bsc Ide Subdvson defnes smooth curve or surfce s the lmt of sequence of successve refnements.
2 D Emles D Rules Effcency Comct Suort Locl Defnton Affne Invrnce Smlcty Contnuty
3 Cubc Slne Subdvson Cubc Subdvson Mtr / 8
4 Egen Anlyss Cubc Slnes Egen Vlues Comlete set of egenvectors ) ( ) ( 4 3 ( ) Egen Anlyss s n egenvector of S wth Invrnce under trnslton: S S S S S * ) * ( ) * ( ) * ( ) * (
5 n Egen Anlyss Alyng subdvson mtr S: S n After lctons: S n Gretest egenvlue cn hve vlue. Egen Anlyss Reetedly lyng the subdvson mtr to set of n control onts results n the control onts convergng to confgurton lgned wth tngent vector.
6 Egen Anlyss Show tht only one egenvlue. Assume two egenvlues. Lmt of the subdvson rocess s the tngent: Tngent s not strght lne > only one egenvlue. lm lm S n Egen Anlyss Smlrly we show tht Mke the orgn then we hve If then > > n n n n
7 Summry The egenvectors should form bss s n egenvector of S wth The frst egenvlue The second egenvlue < All other egenvlues should be less thn Loo Scheme Ordnry verte subdvson
8 Loo Scheme Etrordnry verte nteror verte: vlence other thn 6 boundry verte: vlence other thn 4 Overvew of Subdvson Schemes Vrtonl Subdvson rules chnge bsed on globl energy mnmzton functon Sttonry Aromtng Interoltng Verte Inserton Trngulr Meshes Loo Modfed Butterfly Qudrlterl Meshes Ctmull- Clrk Kobbelt Corner-cuttng Doo-Sbn Mdedge
9 Vrtonl Subdvson Multgrd methods: Fnd such tht E U E S U where E U E Set S E b Vrtonl Subdvson Mnmze bendng energy functonl
10 Vrtonl Subdvson Loo Scheme C contnuous on regulr meshes C contnuous on etrordnry vertces
11 Modfed Butterfly Scheme Interoltng C contnuous Not C on etrordnry vertces k3 k>7 Ctmull-Clrk Scheme Qudrlterl Meshes Aromtng C contnuous on regulr vertces C contnuous on etrordnry vertces
12 Ctumull-Clrk Scheme (cont) Kobbelt Scheme Qudrlterl romtng C contnuous Two ste subdvson
13 Kobbelt Scheme (cont) Doo-Sbn nd Mdedge Schemes Sngle msk for the scheme C contnuous Dsdvntge: no verte corresondence between meshes
14 Doo-Sbn nd Mdedge (cont) Lmtton of Sttonry Subdvson Problems wth curvture contnuty Decrese of smoothness wth vlence Rles Uneven mesh structure
15 Subdvson Surfces n Ger s Gme Ctmull-Clrk Esy to use n estng systems Quds cture symmetres of nturl nd mnmde obects Modfed to llow shr edges Prmetrze subdvson weghts Hybrd subdvson Hybrd Subdvson Infntely shr rules led ~s tmes s n nteger Lner nterolton between s nd s subdvded surfces Followed by smooth rules
16 Smooth Rules f n e n v n n v f f e v e 4 Infntely Shr Creses Shr edge Verte shr edge smooth rule shr edges crese >3 shr edges corner e v e 8 6 k e v e v v v
17 Cloth Smulton Srng-mss energy functonl Use qud grd for wr nd weft drectons of fbrc Energy Functonl Mnmze wr/weft stretch - E ( ) ( s -) * * - Mnmze skew Ed ( 3 4) Es( )Es(3 4) Mnmze bendng long vrtul threds * * * E ( 3) [C( 3) - C( 3)] C( 3) - * * * * 3 3
18 N Collson too slow Use subdvson herrchy Unsubdvde the mesh Mrk ll non-boundry level l edges for mergng Merge fces f f nto f* Remove ll the edges of f* untl ll level l edges hve been merged Buldng the Herrchy Prerocessng ste When verte moves boundng boes re udted bottom u Ech lef onts to t s vertces Test ech verte gnst obect herrchy
19 Teture Mng Assgn smoothly vryng teture coordntes (st) to ll vertces of the orgnl mesh Aly subdvson rules to ( y z s t) Hmm. As lcble to mesh recognton Inherent smlfcton lgorthm
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